Question

If the matrix $\left[\begin{array}{cc}2& 3\\ 5& -1\end{array}\right]=A+B$, where A is symmetric and B is skew symmetric, then B = ______

• A. $\left[\begin{array}{cc}2& 4\\ 4& -1\end{array}\right]$
• B. $\left[\begin{array}{cc}0& -2\\ 2& 0\end{array}\right]$
• C. $\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$
• D. $\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$

Answer 1

Answer: (D)

$\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$

A matrix can be written in the form of a sum of symmetric and skew symmetric matrices, where the skew symmetric matrix can be written in the form $\left(\frac{P-P^T}{2}\right)$ if P is the original matrix.

Similarly, for the above problem, $B=\frac{1}{2}\left[\begin{array}{cc}2& 3\\ 5& -1\end{array}\right]-\frac{1}{2}\left[\begin{array}{cc}2& 5\\ 3& -1\end{array}\right]$

$\therefore B=\frac{1}{2}\left[\begin{array}{cc}2-2& 3-5\\ 5-3& -1+1\end{array}\right]$

$\therefore B=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$